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arXiv:1312.3773 · Physical Review E

Dependence of asymptotic decay exponents on initial condition and the resulting scaling violation

Sourish Bondyopadhyay

Published Dec 12, 2013• 11 citations
arXivHTMLPDF

Abstract

There are several examples which show that the critical exponents can be dependent on the initial condition of the system. In such situations, there are many systems where various issues related to the universal behavior, e.g., the existence of universality, the splitting of the universality class, scaling violations, whether the initial dependence should persist even after a sufficiently long time or is a transient effect, the reasons for such features, etc. are not yet quite clear. In this article, with the simple example of the conserved lattice gas model (CLG), we investigate such issues and clearly show that under certain situations the asymptotic decay exponents are, in fact, dependent on the initial condition of the system. We show that such an effect arises because of the existence of two competing time scales and identify the initial conditions which capture the universal features of the system.

DOI
10.1103/physreve.88.062125

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There are several examples which show that the critical exponents can be dependent on the initial condition of the system. In such situations, there are many systems where various issues related to the universal behavior, e.g., the existence of universality, the splitting of the universality class, scaling violations, whether the initial dependence should persist even after a sufficiently long time or is a transient effect, the reasons for such features, etc. are not yet quite clear. In this article, with the simple example of the conserved lattice gas model (CLG), we investigate such issues and clearly show that under certain situations the asymptotic decay exponents are, in fact, dependent on the initial condition of the system. We show that such an effect arises because of the existence of two competing time scales and identify the initial conditions which capture the universal features of the system.
We present the mapping relation between a one-dimensional conserved lattice gas model and a two species reaction $A+B\ensuremath{\rightarrow}0$. From the kinetics of $A+B\ensuremath{\rightarrow}0$, we show that the anomalous critical decay of an order parameter in the conserved lattice gas model results from finite-size effects induced by the domain structure for random initial conditions where the scaling relation $z={\ensuremath{\nu}}_{\ensuremath{\parallel}}/{\ensuremath{\nu}}_{\ensuremath{\perp}}$ is violated. A simple initial condition satisfying the scaling relation is also discussed.
The conserved lattice gas model in one dimension, generated from various initial states, was intensively studied. The dynamic critical exponents α and ν ║ associated with the decay of activeparticle densities and the correlation time, respectively, were found to depend drastically on the initial states. The perfectly-ordered initial state prepared by repeatedly placing the block ‘0011’ yielded a standard critical behavior, with the critical exponents satisfying all the scaling relations. On the other hand, the natural initial states and the ordered initial states of blocks of odd numbers of particles yielded an exponent α slightly larger than that of an ordered state of ‘0011’, but the data for active-particle densities did not satisfy scaling function. The exponents α and ν ║ were also calculated from, respectively, the autocorrelation function and the persistence distribution of the active-particle densities, both in the steady states. The value of α was found to be close to the value for an ordered state of ‘0011’ whereas the value of ν ║ was consistent with that for the random initial states.
We now begin our exposition of universal dynamic scaling behavior that emerges under non-equilibrium conditions. We are mainly concerned with the relaxational models A and B here, but also touch on other dynamic universality classes. First, we consider non-equilibrium critical relaxation from a disordered initial state, and compute the associated universal scaling exponents within the dynamic renormalization group framework. Related phenomena in the early-time regime, prior to reaching temporally translationally invariant asymptotics, are ‘aging’ and a non-equilibrium fluctuation-dissipation ratio, both of which have enjoyed prominence in the literature on ‘glassy’ kinetics. In this context, we derive the critical initial slip exponent for the order parameter, and briefly discuss interesting persistence properties. Second, we explore the long-time scaling laws in phase ordering kinetics and coarsening for the relaxational models, following a fast temperature ‘quench’ into the ordered phase, in systems with either non-conserved or conserved order parameters. Aside from a few explicit computations in the spherical model limit (wherein the number of order parameter components n → ∞), we largely employ phenomenological considerations and scaling theory, stressing the importance of topological defects for energy dissipation during coarsening. Next we address the question how violations of Einstein's relation that links relaxation coefficients and noise correlations might affect the asymptotic dynamic critical scaling behavior. Whereas in systems with non-conserved order parameter typically the equilibrium scaling laws are recovered in the vicinity of the critical point, in contrast genuinely novel universality classes may emerge in the case of a conserved parameter, provided it is driven out of equilibrium in a spatially anisotropic fashion.
We present very accurate numerical estimates of the time and size dependence of the zero-temperature local persistence in the 2d ferromagnetic Ising model. We show that the effective exponent decays algebraically to an asymptotic value θ that depends upon the initial condition. More precisely, we find that θ takes one universal value 0.199(2) for initial conditions with short-range spatial correlations as in a paramagnetic state, and the value 0.033(1) for initial conditions with the long-range spatial correlations of the critical Ising state. We checked universality by working with a square and a triangular lattice, and by imposing free and periodic boundary conditions. We found that the effective exponent suffers from stronger finite size effects in the former case.
We investigate the global persistence properties of critical systems relaxing from an initial state with non-vanishing value of the order parameter ( e.g. , the magnetization in the Ising model). The persistence probability of the global order parameter displays two consecutive regimes in which it decays algebraically in time with two distinct universal exponents. The associated crossover is controlled by the initial value m 0 of the order parameter and the typical time at which it occurs diverges as m 0 vanishes. Monte Carlo simulations of the two-dimensional Ising model with Glauber dynamics display clearly this crossover. The measured exponent of the ultimate algebraic decay is in rather good agreement with our theoretical predictions for the Ising universality class.
A system driven in the vicinity of its critical point by varying a relevant field in an arbitrary function of time is a generic system that possesses a long relaxation time compared with the driving time scale and thus represents a large class of nonequilibrium systems. For such a manifestly nonlinear nonequilibrium strongly fluctuating system, we show that there exists universal nonequilibrium critical behavior that is incredibly well described by its equilibrium critical properties. A dynamic renormalization-group theory is developed to account for the behavior. The weak driving may give rise to several time scales depending on its form and thus rich nonequilibrium phenomena of various regimes and their crossovers, negative susceptibilities, as well as a violation of fluctuation-dissipation theorem and hysteresis. An initial condition that can be in either equilibrium or nonequilibrium but has longer correlations than the driving scales also results in a unique regime and complicates the situation. The implication of the results on measurement is also discussed. The theory may shed light on the study of other nonequilibrium systems and even nonlinear science.